• Bruce E Oddson added an answer:
    Is it reasonable to use the Hedge d (or g) in case of linear mixed models?

    I would like to compare some studies using the Hedges'd (or its unbiased estimate g). but in both case i have a hierachihcal structure of the data related to transect representing a secondary level o sampling which correlate some of the observation. I was wondering to simply apply an ANCOVA using this transect as a covariate (could it be done?). It would be surely better to use a mixed model with transect as random effect factor, but in this case may I us this Hedge statistics? How could i adjust for the random effect factor?

     I use R, thus lme4 package for mixed models and the compute.es for the effect size

    Bruce E Oddson

    Dear Lorenzo,

        I am not sure why you would obtain "quite different" results. They should be very close to the same. In totally balanced designs and typical linear model I expect them to be the same. If you have covariates and unbalanced n's then they should be "near" the model specification. Both of these assume that the model specification is sound - so maybe this is telling you about a problem.

    How far different is the SD from the two methods? Do you have the sort of joint distributions and homoscedastic residuals expected for all the variables involved?

    I might start to debug by redoing with the simplest model and its marginal means, then put in 1 level 2 factor and redo, etc.. 

    Good luck

  • Ahmad Rayan added an answer:
    Alternative for ANCOVA?

    If the homogeneity of regression slopes assumption for ANCOVA (no interaction between the covariate and the independent variable) was violated, what is the next step to perform the analysis. Is there any alternative test for ANCOVA?

    For instance, you want to use analysis of covariance (ANCOVA), with post-test scores as dependent, pre-test scores as covariates, and group membership as independent factor.

    Ahmad Rayan

    Highly appreciated dear Bruce Weaver. Hope you the best...

    Regards, Ahmad.

  • Parto Yazdani added an answer:
    Ancova, anova or paired t test?

    Hi everyone,

    I am puzzled in what test to use when tetsing if two sample outcomes differ statistically significant. Should I use the t test, ancova or anova?In what cases should I consider anova or ancova above t test? I almost always use t test, and also tell my students to do so. Who can help me out?



    Parto Yazdani


    I was wondering if I can use ANCOVA for two groups? In other words, how do you control for a factor in a t test comparison?

    Thank you!

  • Thom S Baguley added an answer:
    Can we do ANCOVA for single group pretest post design assuming the pre test as covariate?

    I did an experimental intervention study based on single group pretest post test design. A pre test followed by a post test after intervention. I also conducted a delayed post test.Is it possible to calculate ANCOVA using spss by assuming pre test as covariate?

    Thom S Baguley

    I don't see a problem with having the two delays as repeated measures (but advise centering the covariate beforehand). Generally having the pre-test as covariate rather than a repeated measure produces a more plausible model and has greater statistical power.

  • Renaud Lancelot added an answer:
    Can I use ANOVA and ANCOVA repeated measures although scores are not normally distributed?

    1. Can I consider the both tests have been robost by violating the normality procedures? Can I still use the both tests? Will it be valid?

    Renaud Lancelot

    This statement is confusing. Many parametric models don't need a normality assumption, and when relevant, this normality assumption is only needed for residuals, when testing model coefficients.

  • Nan Mogean added an answer:
    Respected Researchers, can I integrate parametric and nonparametric test?

    I have been using the mann whitney and kruskal wallis test as the scores are not normally distributed. I have compared two different groups in the study. This serves as non-parametric research. However, I have also conducted ANOVA repeated measures and ANCOVA repeated measures in order to identify the effect in within the sessions. Can I refer to parametric test basd on the absolute scores values although the scores are not normally distributed?

    Nan Mogean

     Respected researchers,

    How if my degree of freedom is lower then 40? Can I still stick to the ANOVA and ANCOVA?

  • Nan Mogean added an answer:
    Can I use these both parametric tests although my scores are not normally distributed please?

    Based on my scores, they are not normally distributed, Thus, I have conducted Friedman Test and also ANOVA and ANCOVA repeated measures based on the absolute means scores in order to identify changes within five sessions in a group. Can I use these both parametric tests although my scores are not normally distributed please?

    Nan Mogean

    1. the total number of correct answers are known and within the story.

    2. no, the answers differ for each session as the stories used are different 

    3. the answers were free

    4. if the answers are free, the correct answers derived from the actual correct proposition recalled by the students like in the story. we have developed a list of idea units.

  • Patricia Rodriguez de Gil added an answer:
    Can you help me with reporting effect size vis-a-vis partial eta squared?


    I am doing experimental work in consumer psychology (2 X 2 design), and doing my analysis using standard ANOVA/ ANCOVA. My question is about reporting effect size vis-à-vis partial eta squared obtained from SPSS results. All papers I’ve seen refer to Cohen’s (1988) book. However, I’ve not come across any reference which outlines the effect size (weak, medium, strong) corresponding to the range of partial eta squared. Lakens (2013) discusses effect size reporting, including partial eta squared in detail, but there is no guide of which level of partial eta squared corresponds to what effect size. Can you help me with this?

    Works Cited:

    Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences. New York, NY: Routledge Academic

    Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4(3), 1–12. http://doi.org/10.3389/fpsyg.2013.00863

    Patricia Rodriguez de Gil

    Daniel, I add my sentiments to Diptiman's! Thank you so much for sharing your file!

  • John Christie added an answer:
    What is a test for the homogeneity of slopes before running an ANCOVA? What is the meaning of an interaction term in R?

    I have some doubts in how to interpret the interaction term in an ANCOVA using R since in other statistical programs this term is not provided.
    If I am right, one assumption of ANCOVA is the Homogeneity of regression slopes (that is, they must be parallel). Not following this assumption means you cannot use ANCOVA.

    My question is, can the interaction term be interpreted as a test for the homogeneity of regression slopes? I meant, if is significance is because the slopes are not parallel and the model cannot be ran.
    If not, how can I test for it?

    Here is an example
    where weight and size are continuous and sex is dummy variable

    > ancova<-lm(log(weight)~sex*log(size))
    > anova(ancova)
    Df Sum Sq Mean Sq F value Pr(>F)
    sex 2 31.859 15.9294 803.9843 <2e-16 ***
    log(weight) 1 11.389 11.3887 574.8081 <2e-16 ***
    sex:log(weight) 2 0.063 0.0317 1.6021 0.2025


    John Christie

    They're both tests of interaction. With R you set it up yourself so in order to make it match you need to look at the Statistica manual.

  • Irena Pavela added an answer:
    How do you conduct a mixed-factors ANCOVA with a time-dependent covariate in R?

    This might be a long shot, but I thought I'd give ResearchGate a chance to prove itself.

    I'm having trouble trying to conduct an ANCOVA in R when one of my variables is a time-dependent covariate. For simplicity sake, let's say I have variables Y, A, B, and X, where Y is my dependent variable, A is a between subjects factor with two levels, B is a within-subjects factor with 6 levels, and X is a continuous variable I want to add as a covariate and is measured at all levels of A and B.

    Any help on how to conduct this in R using either lm(), lme(), aov(), ezANOVA(), or something similar would be very helpful.


    Irena Pavela

    Dear Scott, thank you for your prompt reply and book recommendation. 

  • Ciro Cabal added an answer:
    May I use covariates that are known correlated with the factor when carrying out an ANCOVA?

    Hello. I was wondering if it is correct to use covariates that one knows to be directly related to one's factor, in order to improve the results of an ANCOVA. For example, in a nitrogen deposition experiment in which N addition is the factor, NO3- and NH4+ in the soil before fertilization can be considered covariates, ok. But, is it acceptable as well to measure NH4+ and NO3- when collecting samples of dependent variables, after fertilizing, and include those measures as covariates of the N treatment factor in an ANCOVA? I guess it is not but I have seen it published in good journals... Thank you!

    Ciro Cabal

    ANOVA (Factor: N fertilization treatment - Dependent var: Twigs growth)

                        Df     Sum Sq       Mean Sq         F value  Pr(>F)
    Nt                1       0.000213    0.0002132     0.547      0.465
    Residuals  34     0.013257    0.0003899


    MANOVA (Factor: N fertilization treatment - Dependent var: Twigs growth + [N] in NO3- + [N] in NH4+): The response variable that is of interest still have the same p-value.

                         Df      Pillai       approx F  num Df   den Df       Pr(>F)
    RomP$Nt    1      0.28599    4.2725              3           32     0.01205 *
    Residuals   34

    Response 1 : Twigs growth
                         Df       Sum Sq       Mean Sq       F value       Pr(>F)
    RomP$Nt   1    0.0002132   0.00021318      0.5467      0.4647
    Residuals  34  0.0132573   0.00038992

    Response 2 : [N] in NH4+
                         Df       Sum Sq       Mean Sq        F value       Pr(>F)
    RomP$Nt   1          322.82         322.82          8.1135    0.007405 **
    Residuals  34       1352.80       39.79

    Response 3 : [N] in NO3- 
                         Df       Sum Sq       Mean Sq       F value       Pr(>F)
    RomP$Nt   1          68.911            68.911       11.816    0.001568 **
    Residuals  34       198.289             5.832
    Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


    ANCOVA (Covariates: [N] in NO3- + [N] in NH4+ - Factor: N fertilization treatment - Dependent var: Twigs growth)

    (Intercept)     NNO3               NNH4              Nt
    0.0746213   0.0011759   -0.0007595   0.0001813

    Response: Twigs rowth
                       Df       Sum Sq        Mean Sq          F value       Pr(>F)
    NNO3         1     0.0000486    0.00004861    0.1218         0.7293
    NNH4         1     0.0004051    0.00040506    1.0152         0.3212
    Nt                 1     0.0002488    0.00024877     0.6235         0.4356
    Residuals 32     0.0127680    0.00039900


    In no case my response variable significantly variates with the factor, but this may just mean that N deposition has no effect on it. However, the experiment is a basic block design and the variability between blocks is higher than within blocks, thus maybe I can try a Mixed Linear Model. Moreover, my question is still theoretical and not necessarily applied to those data: May I conclude that NNO3- measured after fertilization (which is a response variable) could be considered as covariate as well if it is not highly correlated to the treatment? I mean, the theoretical independence of factor vs covariates, despite the real interaction that we can measure, is not an assumption to ANCOVA?

    Thank you again!

  • Raid Amin added an answer:
    I have a very large Mean Square Error (10307) in an ANCOVA. The result is significant (p=.036). The data were transformed; is the large MSE a problem?

    I performed a one-way ANOVA on transformed data (to correct for not normal distribution) covarying one variable (age). The result is significant, p=.036, but there is a huge Mean Square Error of 10307. I understand that the MSE should be small. Is this large MSE a problem?

    Raid Amin

    Sandro: I am trying to help Graham to better understand why he has an MSE that seems to be large. If we knew more about the transformation used, this could shed some light on the magnitude of the resulting values, and then we could better understand if it was a poor modeling or simply a result of using large values in the analysis, which then would not be a reason for concern for him.

  • Abdolghani Abdollahimohammad added an answer:
    When reporting ANCOVA, what are the appropriate descriptives?

    I am doing a pretest-posttest study, controlling for pre-scores (cov) in the analyses of post-scores (DV) by intervention type (IV).

    I understand that significant ANCOVA should be interpreted using the adjusted means, so would this be the right thing to report alongside the ANCOVA result? (We are expected to have a table of descriptives for all groups, including where nothing significant has been found, along with our inferential stats). And is there an 'adjusted' equivalent of standard deviation or does this stay the same?

    Additionally, I've been recommended to use Cohen's d when I come to report effect size - how does this work with ANCOVA - do I use the adjusted means to calculate d? And the original standard deviation?

    Would greatly appreciate any help, or redirecting to a clear example of ANCOVA reporting!

    Thanks =)


    Abdolghani Abdollahimohammad

    You should report Mean (confidence interval) of each group, mean difference (95% CI of mean differences), and P-value. An example:

    Groups        Mean (95%CI)              Mean difference (95%CI)                P-value


    A                 23.12 (12.11, 27.87)


                                                                -4.24 (0.23, 1.54)                      0.042

    B                27.36 ( 23.02, 28.14)

  • Abdulrazzak Charbaji added an answer:
    I just ran an ANCOVA model in which I got a positive sign for constant and a negative for the slopes of dummies variable, how do I interpret this?

    For a multiple dummy variable model without an interaction term

    Details are given below. 

    A pozitive sign for constant (bo=6.68) and a negative for slopes (b1=-6.60, b2=-6.75, b3=-6.90, b4=-6.80)?

    Abdulrazzak Charbaji

    Using Dummies in analysis of covariance helps for control (removing the effect of covariate) keeping in mind that the coefficient of the dummy variable test the shift in intercept (up or down) and the coefficient of the interaction measures the change in slope  You may check this reference:

    Applied Econometrics using Eviews, SPSS & Excel with Applications in Arab Countries (2012). Available at AMAZON:

  • Bruce Weaver added an answer:
    What can I do while doing two group multivariate analyses with unequal group sizes?

    I want to compare performances of 9 patients with those of 45 healthy matched controls on a series of cognitive tests. Most of the related studies have used ANOVA, even when the comparisons have been between two groups on only one dependent variable. But, all the studies have had relatively equal group sizes (8-8, 9-9, 9-10, 11-12, etc.). What can I do with the large difference between the group sizes while doing one-way ANOVA, mixed ANOVA (group[patient, normal] as between subject variable, subtests/conditions as within-subject variable, and a dependent variable [e.g., number of correct responses]), or ANCOVA (baseline tests as covariates)? Meanwhile, Levene's Test shows no deviations from homogeneity of variance (homoscedasticity) for most of the variables (not for all), also, the p values (for differences between the groups) are very low.

    Bruce Weaver

    I wonder what would happen if Fagerland & Sandvik's simulations were performed using Swedish or Finnish computers.  Would the results change?  :-)

  • Rainer Duesing added an answer:
    What is the best way to analyze a 2x4 within design, with a metric between variable?

    I have a 2 (valence: pos vs neg) x 4 (stimulus type) design. Both variables were assessed within each subject. Additionally, I have a metric trait variable. A simple way would be to do a median split an incorporate it as a between variable to a 2x2x4 ANOVA, but I am aware of problems with median splits (although sometimes exaggerated, see DeCoster, Iselin, & Gallucci, 2009).

    I tried to run the it as an ANCOVA, including the trait variable as a metric variable, but I am not really satisfied with the interpretability of this option.

    Do you have any other ideas how to analyze the data and keeping the trait variable as a metric one?

    I use SPSS and MATLAB.

    Rainer Duesing

    Yes, I have the Tabachnick & Fidell book (great book) at hand but ignored this chapter too long I suspect. I also have West, Welch, and Galecki - Linear mixed models. Roh, I thought if you already used R, you could give a quick introduction ;-)

    I think we both attended to the same MPlus workshop. Maybe I'll ask Timo

  • Prasanth Sasidharan added an answer:
    What is the difference between ANCOVA and Repeated measure ANCOVA?

    Could you please guide me on ANCOVA and repeated measure ANCOVA? What is the difference between these two, Please guide using an example.

    Prasanth Sasidharan

    Hi all Thanks for the help

    I have few more clarity required like  ANCOVA and Repeated measure Analysis of covariance not ANOVA. Plz suggest 

  • Amber Muhinyi added an answer:
    Can anyone advise on how I can conduct a power analysis for ANCOVA using GPower or another method?

    Gpower requires that df numerators be specified - can anyone advise how to estimate these in order to determine sample size? Thank you

    Amber Muhinyi

    Thank you Joan - this looks excellent.

  • Massimiliano Grassi added an answer:
    Is wild bootstrap valid in a “bordeline" case t-test?

    I have to compare the mean levels of a continuos variable y (ranging 1-20) subdividing my sample in two groups according to a dichotomous variables (i.e gender). Sample sizes of the two groups are unequal (m=20/f=80).

    It happened that in the male (smaller) group, all subjects have y=1, while in the female group scores ranges along all possible scores, although the distribution is not normal and highly skewed towards the lower scores. As of these, the male subgroup has no variance in y while the female group shows it.

    Considering these issues, I thought to use bootstrap on t-test to make a more reliable mean comparison between the two groups: 

    I first applied bootstrap, stratified for gender, with (welch) t-test. 
    I then tried a wild bootstrap approach with the same t-test. As a matter of fact, I can consider this also a special regression with a single dichotomous predictor, with an extreme hetehroscedasticy and non-normality of residuals and in regression with heteroscedasticy the wild bootstrap approach is usually recommended.
    What relevantly differs between the two bootstrap strategies is that with a wild approach I’m bootstrapping residuals from one group to subjects of another group. Results are also very different: wild bootstrap provides much much smaller p/CI than the stratified bootstrap with welch correction.

    My questions are:

    1 is a boostrapped t-test valid in this situation? And if so,
    2 what of the two bootstrap approach is the most correct in this situation?
    3 in case I would add some covariates (i.e. ANCOVA with one dichotomous predictor and two continuos predictors), what is again the most correct bootstrap strategy?

    Many thanks for your help!

    Massimiliano Grassi

    Thank you all and sorry for the late reply (notifications went to spam folder...)!

    I totally agree with you all and as you got it, my worries rely on possible reviewers' concerns on not providing a formal testing for the group difference. I think beyond the explanation I can add a bootstrapped CI of the female mean to show it does not include the 1, that seems to me a easy-to-be accepted compromise.

    Actually I have a couple of more y’s to test with the same model (showing heteroskedasticy and non-normality of residuals, even though non linearity seems not to be the issue) and I think i will go for a wild bootstrap for these.

    And thank you all for the general reminders that are truly valid for every analysis!

  • Jochen Wilhelm added an answer:
    Can anybody help on repeated measure ANCOVA?

    Is it possible to do R ANCOVA with only pre and post values and pre value as covariate and time vs treatment interaction? Plz help.........

    Jochen Wilhelm

    Just to unreval the common concept:

    The example in Gustavo's link proposes to calculate the repeated-measures-ANOVA as

    summary(aov(price ~ store + Error(subject/store), data=groceries2))

    The very same same is obtained from the anova of the linear model

    anova( lm(price ~ subject + store , data=groceries2) )

    what is of the form I proposed above.

    The difference is that the aov-function gives SumSq values between subjects within stores, whereas the lm-function gives the SumSq between subjects over all stores. If this is not the interesting point, both ways eare equivalent, since they calculate the same resut for the effect of "strore", the lm with the advantage that the actual differences between the stores can be more directly infered.

    Both methods also work for unbalanced designs. However, aov thows a warning that the Error() model is singular (deleting some entries causes some stores have no measurements for some subjects and "Error(subject/store)" can not be estimated there; the "problem" can be "solved" by again allowing the between.subject error be calculated over all stores: "Error(subject)")

  • Meg Barber added an answer:
    How can one account for confounding variables in data analysis?

    I am hoping somebody can help me. I am currently writing my protocol for my research module, however the data analysis section is baffling me! 

    My study is an RCT with two treatment arms, and I am wondering which is the best method for accounting for/controlling potential confounding covariates. I have visited the idea of performing within-group ANCOVA, and then further statistical analysis based on the F-test significance (e.g. if significant- create subsets and analyse with ANOVA or multiple student t-tests for continuous endpoints) and if F-test finds no significant difference between covariates, just to analyse as originally intended (e.g. student t-test for continuous and chi square for categorical). I have also considered Bonferroni correction but I am unsure if this would be suitable. 

    Apologies for rambling- my mind is swimming! Many thanks for any advice to be offered!

    Meg Barber

    That's brilliant, thank you for your help!


  • Roshanak Soltani added an answer:
    Any suggestion about using ANCOVA with repeated measures?

    My consulting adviser said that we can't use covariance method when there are more than 2 time points. But I'm not sure about it again!

    What's your idea about that?

    Roshanak Soltani

    Dear Zenhng,

    Unfortunately no, not yet.

  • Marina Menez added an answer:
    How should I analyze results from the study that involved pre- and post-test measurements and had one experimental and one control group?

    Should I go for ANCOVA (use “group” as fixed factor, and pre-test as covariate)? Or should I go for Split-plot ANOVA (use “group” as between-subjects factor and measurement occasion as within-subjects factor) and look for significant interaction?

    Marina Menez

    Ok. Thanks! 

    (Estimated marginal) mean differences in ANCOVA could also could go into different directions than I expect.

    I agree, but given you are already controlled by possible previous differences between your groups you are in a better posibility to establish a) the effect was significant b) the direction of difference is in the direction your test indicates. You can´t assert that in the mixed ANOVA (well, not in a direct way; you should run other tests like Tukey etc.).

    Try  to get Judd and mcClelland's book...it has all this stuff, presented in a very friendly way...the other good reference is Maxwell and Delaney's book, Designing experiments and analyzing data but it's a little more technical.

  • Haris Memisevic added an answer:
    When should I perform ANCOVA?

    I was wondering about when is it appropriate to use covariate in the analysis of data. Some cases are straightforward such as when you have pre – post intervention scores in homogeneous groups and use pre intervention scores as a covariate.

    But, what to do in the case when comparing for example, motor skills in people with intellectual disability and those without IT. Does it make any sense to include IQ as a covariate when we know that these two groups are different in IQ scores? Any thoughts on this?

    Haris Memisevic

    Reading the comments I finally realized the right question. I guess you should not use (or at least be very careful about using) covariate when the groups are naturally different on the covariate (not randomly assigned to the group). Thank you for clarifying this concept.

  • Moath Awawdeh added an answer:
    Can anyone recommend a good book which studies the One-Way Analysis of Covariance (ANOCOVA or ANCOVA) with applications and tables interpretation ?

    I am using ANOCOVA for both of its goals (regression and grouping) using Matlab programming (aoctool). I used Schewart's lectures note and Huck's book (Reading statistic and research) as references but I need more deep studies which present this analysis technique in more detail with more engineering applications, unfortunately I don't use SPSS!  

    Thanks all  

    Moath Awawdeh

    Thank u all!

  • Mikhail Saltychev added an answer:
    How do I use ANCOVA for meta-analysis?

    Trying to conduct a meta-analysis using post-test values from several studies with two independent groups (cases/controls). There are pre-/post-test means and SDs. What I'm looking for, are tips on how to employ ANCOVA for adjusting for difference between baseline data. What software should I use? I'm quite familiar with CMA and MIX. I'm not sure, but I think that CMA is converting pre-/post-values into change difference. I understand that it works too, but, in this particular case, I'd like to try ANCOVA. Please, do not suggest R. I just don't want to spend too much time on learning the new language. Unfortunately, I do not also have Stata at my disposal. If it is too complex, than I just stick with change difference. Sorry if I couldn't express myself more clearly.

    Mikhail Saltychev

    Thank you Tom. I think, meta-regression is not what Cochrane Handbook means when talking about ANCOVA. I did a little search on Internet and, hopefully, found the answer. It seems, that you have to calculate (manually) an effect size for each included study using ANCOVA, and then to put them into meta-analysis (any software, CMA is just fine). There is a presentation on Cochrane site by Jo McKenzie.

    Andrew, propensity score is hardly of use in meta-analysis, except maybe for IPD. PS needs huge numbers. I did my PhD using propensity score on sample of 50 000. I believe, there is no way you can use propensity score in a meta-analysis on aggregate scores. 

  • David MacKinnon added an answer:
    How do I perform a mediation analysis using pretest and post-test scores?

    I have a between-subjects manipulation that has 2 levels (X). I want to see if this manipulation affects a performance score, Y. I also want to see a variable, which is measured before Y, would mediate the X on Y effect. So this is the classic mediation problem: X --> M --> Y

    I measured both M and Y before and after the manipulation, so that I have M1, M2, and Y1, Y2. My research makes more sense in whether the change in M (i.e. M2-M1) explains the change in Y (i.e. Y2-Y1). A friend suggested me to do the mediation with the difference scores:

    X --> (M2-M1) --> (Y2-Y1)

    I'm also considering controlling the M1 score and Y1 score in building the mediation model, which should work like an ANCOVA when M1 and Y1 don't interact with X in influencing Y2.

    Appreciate your help if you could let me know which way is a better approach in this question.

    PS Also considered multilevel and SEM approach, but not quite feasible with my small sample size. That's why I wanna stick with the regression approach.

    David MacKinnon

    Hello Henry,

    See Chapter 8 in MacKinnnon (2008, Introduction to Statistical Mediation Analysis) for options with two waves of data and also options for more than two waves of data. There are several options with the pretest-posttest design including difference score, residualized change score, and analysis of covariance. Residualized change and ANCOVA general give very similar results except if there are substantial pre-test differences. The choice depends on the pattern of change over time that you would expect in each group if no intervention was delivered. If the groups were randomized, differences between groups are likely due to random error so regression to the mean would be expected over time--so ANCOVA and residualized change would be reasonable. If the groups are not randomized, a difference score method may be more appropriate as it allows for pretest differences to maintain over time. The ANCOVA model is the most general and it can be estimated with regression or SEM. The SEM approach is more general as it can allow for more complicated models such as models with latent variables and extensions to more waves of data. These issues are described in MacKinnon (2008) as mentioned above. The two wave ANCOVA model and multi-wave longitudinal mediation models are described in MacKinnon (1994: NIDA monograph) and applied to the mediation analysis of a steroid prevention program in MacKinnon et al. (1991; Prevention Science). I believe that both of these papers are on Research Gate. If you can’t get them on Research Gate, please contact me and I will send them to you.
    Dave MacKinnon

    MacKinnon, D. P. (1994). Analysis of mediating variables in prevention and intervention research. In A. Cazares & L. A. Beatty (Eds.), Scientific methods for prevention/intervention research (NIDA Research Monograph Series 139, DHHS Pub 94-3631, pp. 127-153). Washington, DC: U. S. Department of Health and Human Services.
    MacKinnon, D. P. (2008). Introduction to statistical mediation analysis. Mahwah, NJ: Erlbaum.
    MacKinnon, D. P., Goldberg, L., Clarke, G. N., Elliot, D. L., Cheong, J., Lapin, A., et al. (2001). Mediating mechanisms in a program to reduce intentions to use anabolic steroids and improve exercise self-efficacy and dietary behavior. Prevention Science, 2, 15-28.

  • Robert J Miller added an answer:
    How can I compare linear relationships?

    We have measured simple relationships between the size of different species and their weight (biomass).  They are linear regressions (sometimes semilog). The type of question I would like to answer is, for example, if I have 4 species of snail, each with a separate linear equation representing the relationship between size and biomass, are those relationships significantly different, or does one relationship suffice for all my snail species?  

    I'm not sure how to do this comparison. Each species often has different sample sizes. I thought about just doing the separate relationships, then pooling all samples and comparing the resulting slope to the species-wise relationships with t tests. However I'm worried that species with more samples will bias the results. I could randomly remove samples from those species to equalize them. Another alternative might be to do an ancova with species put in as a dummy variable and look for interactions with species as a test of parallelism. Does that seem reasonable?  It seems like a good idea to me because it will also be a test of whether the intercepts are the same.

    Robert J Miller

    Excellent points Timothy, thanks for your input.

  • Bruce E Oddson added an answer:
    Does anyone have suggestions for reporting a robust ANCOVA?
    I'm following the example in Andy Field's R book where he suggests that after failing the test for homogeneity of regression slopes, one might do a robust ANCOVA ala Wilcox 2005. I'm able to run the tests no problem, and interpreting them is also not an issue, but for output of the following nature (see below), does anyone know of a standard way to report this data?

    I think a way to start at least will be to report the standard ANCOVA up to the point where the interaction is significant and then say robust procedures were followed, how to report these though are a bit beyond me.

    ancova(covGrp1, dvGrp1, covGrp2, dvGrp2)
    [1] "NOTE: Confidence intervals are adjusted to control the probability"
    [1] "of at least one Type I error."
    [1] "But p-values are not"
    X n1 n2 DIF TEST se ci.low ci.hi p.value crit.val
    [1,] 10.30 20 12 -22.166667 2.7863062 7.955575 -47.42320 3.089867 0.0213100575 3.174696
    [2,] 11.30 28 17 -19.184343 2.7536447 6.966891 -39.98396 1.615273 0.0167914292 2.985495
    [3,] 12.45 32 23 -20.350000 3.9162704 5.196270 -35.02758 -5.672423 0.0008787346 2.824637
    [4,] 14.00 27 34 -8.314171 1.4638404 5.679698 -23.71193 7.083583 0.1524122220 2.711016
    [5,] 16.10 14 17 3.431818 0.3796813 9.038682 -22.28197 29.145604 0.7085490133 2.844860

    ancboot(covGrp1, dvGrp1, covGrp2, dvGrp2,tr = .2, nboot=2000)
    [1] "Note: confidence intervals are adjusted to control FWE"
    [1] "But p-values are not adjusted to control FWE"
    [1] "Taking bootstrap samples. Please wait."
    X n1 n2 DIF TEST ci.low ci.hi p.value
    [1,] 10.30 20 12 -22.166667 -2.7863062 -47.00379 2.670459 0.0355
    [2,] 11.30 28 17 -19.184343 -2.7536447 -40.93482 2.566135 0.0185
    [3,] 12.45 32 23 -20.350000 -3.9162704 -36.57264 -4.127360 0.0015
    [4,] 14.00 27 34 -8.314171 -1.4638404 -26.04606 9.417719 0.1525
    [5,] 16.10 14 17 3.431818 0.3796813 -24.78674 31.650380 0.6980
    Bruce E Oddson

    Dear John,

       If you are going to be fair (depends how you look at it) to other robust techniques, then I would say you report it as simply as a regular ANCOVA. You state which package and assumptions you used. You give the p values and associated CIs for each statistic of interest. Although the additional information provided by the procedure is potentially helpful, nobody asks for it when (often incorrectly) "standard" procedures are used. 

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